SSC/Structure/LimitingMasses - Revision history

Jet53man: /* Schönberg-Chandrasekhar Mass */

2023-11-17T17:32:00Z

Schönberg-Chandrasekhar Mass

← Older revision		Revision as of 13:32, 17 November 2023
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	In the early 1940s, Chandrasekhar and his colleagues (see [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] and [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]) discovered that a star with an isothermal core will become unstable if the fractional mass of the core is above some limiting value. They discovered this by constructing models that are now commonly referred to as ''composite polytropes'' or ''bipolytropes'', that is, models in which the star's core is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the star's envelope is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. In [[SSC/Structure/BiPolytropes#BiPolytropes\|an accompanying discussion]] we explain in detail how the two structural components with different polytropic indexes are pieced together mathematically to build equilibrium bipolytropes. For a given choice of the two indexes, <math>~n_c</math> and <math>~n_e</math>, a sequence of models can be generated by varying the radial location at which the interface between the core and envelope occurs. As the interface location is varied, the relative amount of mass enclosed inside the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, quite naturally varies as well.		In the early 1940s, Chandrasekhar and his colleagues (see [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] and [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]) discovered that a star with an isothermal core will become unstable if the fractional mass of the core is above some limiting value. They discovered this by constructing models that are now commonly referred to as ''composite polytropes'' or ''bipolytropes'', that is, models in which the star's core is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the star's envelope is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. In [[SSC/Structure/BiPolytropes#BiPolytropes\|an accompanying discussion]] we explain in detail how the two structural components with different polytropic indexes are pieced together mathematically to build equilibrium bipolytropes. For a given choice of the two indexes, <math>~n_c</math> and <math>~n_e</math>, a sequence of models can be generated by varying the radial location at which the interface between the core and envelope occurs. As the interface location is varied, the relative amount of mass enclosed inside the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, quite naturally varies as well.

	[http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] built structures of uniform composition having an isothermal core (<math>~n_c = \infty</math>) and an <math>~n_e = 3/2</math> polytropic envelope and found that equilibrium models exist only for values of <math>~\nu \le \nu_\mathrm{max} \approx 0.35</math>. [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] extended this analysis to include structures in which the mean molecular weight of the gas changes discontinuously across the interface. Specifically, they used the same values of <math>~n_c</math> and <math>~n_e</math> as Henrich & Chandrasekhar, but they constructed models in which the ratio of the molecular weight in the core to the molecular weight in the envelope is <math>~\mu_c/\mu_e = 2</math>. This was done to more realistically represent stars as they evolve off the main sequence; they have inert, isothermal helium cores and envelopes that are rich in hydrogen. Note that introducing a discontinuous drop in the mean molecular weight at the core-envelope interface also introduces a discontinuous drop in the gas density across the interface. As the following excerpt from p. 168 of their article summarizes, in these models, [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] found that <math>~\nu_\mathrm{max} \approx 0.101</math>. This is commonly referred to as the Schönberg-Chandrasekhar mass limit, although it was Henrich & Chandrasekhar who were the first to identify the instability.		[http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] built structures of uniform composition having an isothermal core (<math>~n_c = \infty</math>) and an <math>n_e = 3/2</math> <font color="red">[should be n<sub>e</sub> = 3]</font> polytropic envelope and found that equilibrium models exist only for values of <math>~\nu \le \nu_\mathrm{max} \approx 0.35</math>. [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] extended this analysis to include structures in which the mean molecular weight of the gas changes discontinuously across the interface. Specifically, they used the same values of <math>~n_c</math> and <math>~n_e</math> as Henrich & Chandrasekhar, but they constructed models in which the ratio of the molecular weight in the core to the molecular weight in the envelope is <math>~\mu_c/\mu_e = 2</math>. This was done to more realistically represent stars as they evolve off the main sequence; they have inert, isothermal helium cores and envelopes that are rich in hydrogen. Note that introducing a discontinuous drop in the mean molecular weight at the core-envelope interface also introduces a discontinuous drop in the gas density across the interface. As the following excerpt from p. 168 of their article summarizes, in these models, [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] found that <math>~\nu_\mathrm{max} \approx 0.101</math>. This is commonly referred to as the Schönberg-Chandrasekhar mass limit, although it was Henrich & Chandrasekhar who were the first to identify the instability.
	<div align="center">		<div align="center">
	<table border="1" cellpadding="5" width="60%">		<table border="1" cellpadding="5" width="60%">

Jet53man: /* Schönberg-Chandrasekhar Mass */

2021-07-29T21:47:41Z

Schönberg-Chandrasekhar Mass

← Older revision		Revision as of 17:47, 29 July 2021
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	! style="height: 125px; width: 125px; background-color:#ffeeee;" \|		! style="height: 125px; width: 125px; background-color:#ffeeee;" \|
	<font size="-1">[[H_BookTiledMenu#~~Equilibrium_Structures~~\|<b>Schönberg-<br />Chandrasekhar<br />Mass</b>]]<br /> (1942)</font>		<font size="-1">[[H_BookTiledMenu#MoreModels\|<b>Schönberg-<br />Chandrasekhar<br />Mass</b>]]<br /> (1942)</font>
	\|}		\|}
	In the early 1940s, Chandrasekhar and his colleagues (see [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] and [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]) discovered that a star with an isothermal core will become unstable if the fractional mass of the core is above some limiting value. They discovered this by constructing models that are now commonly referred to as ''composite polytropes'' or ''bipolytropes'', that is, models in which the star's core is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the star's envelope is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. In [[SSC/Structure/BiPolytropes#BiPolytropes\|an accompanying discussion]] we explain in detail how the two structural components with different polytropic indexes are pieced together mathematically to build equilibrium bipolytropes. For a given choice of the two indexes, <math>~n_c</math> and <math>~n_e</math>, a sequence of models can be generated by varying the radial location at which the interface between the core and envelope occurs. As the interface location is varied, the relative amount of mass enclosed inside the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, quite naturally varies as well.		In the early 1940s, Chandrasekhar and his colleagues (see [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] and [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]) discovered that a star with an isothermal core will become unstable if the fractional mass of the core is above some limiting value. They discovered this by constructing models that are now commonly referred to as ''composite polytropes'' or ''bipolytropes'', that is, models in which the star's core is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the star's envelope is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. In [[SSC/Structure/BiPolytropes#BiPolytropes\|an accompanying discussion]] we explain in detail how the two structural components with different polytropic indexes are pieced together mathematically to build equilibrium bipolytropes. For a given choice of the two indexes, <math>~n_c</math> and <math>~n_e</math>, a sequence of models can be generated by varying the radial location at which the interface between the core and envelope occurs. As the interface location is varied, the relative amount of mass enclosed inside the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, quite naturally varies as well.

Jet53man: /* Schönberg-Chandrasekhar Mass */

2021-07-28T20:53:40Z

Schönberg-Chandrasekhar Mass

← Older revision		Revision as of 16:53, 28 July 2021
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	==Schönberg-Chandrasekhar Mass==		==Schönberg-Chandrasekhar Mass==
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:#ffeeee;" \|
			<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>Schönberg-<br />Chandrasekhar<br />Mass</b>]]<br /> (1942)</font>
			\|}
	In the early 1940s, Chandrasekhar and his colleagues (see [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] and [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]) discovered that a star with an isothermal core will become unstable if the fractional mass of the core is above some limiting value. They discovered this by constructing models that are now commonly referred to as ''composite polytropes'' or ''bipolytropes'', that is, models in which the star's core is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the star's envelope is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. In [[SSC/Structure/BiPolytropes#BiPolytropes\|an accompanying discussion]] we explain in detail how the two structural components with different polytropic indexes are pieced together mathematically to build equilibrium bipolytropes. For a given choice of the two indexes, <math>~n_c</math> and <math>~n_e</math>, a sequence of models can be generated by varying the radial location at which the interface between the core and envelope occurs. As the interface location is varied, the relative amount of mass enclosed inside the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, quite naturally varies as well.		In the early 1940s, Chandrasekhar and his colleagues (see [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] and [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]) discovered that a star with an isothermal core will become unstable if the fractional mass of the core is above some limiting value. They discovered this by constructing models that are now commonly referred to as ''composite polytropes'' or ''bipolytropes'', that is, models in which the star's core is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the star's envelope is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. In [[SSC/Structure/BiPolytropes#BiPolytropes\|an accompanying discussion]] we explain in detail how the two structural components with different polytropic indexes are pieced together mathematically to build equilibrium bipolytropes. For a given choice of the two indexes, <math>~n_c</math> and <math>~n_e</math>, a sequence of models can be generated by varying the radial location at which the interface between the core and envelope occurs. As the interface location is varied, the relative amount of mass enclosed inside the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, quite naturally varies as well.

Jet53man: /* Related Discussions */

2021-07-27T22:44:46Z

Related Discussions

← Older revision		Revision as of 18:44, 27 July 2021
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	* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]		* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]
	* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]		* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]
	* [[SSC/Structure/BiPolytropes/~~Analytic51~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]		* [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]
	* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]		* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]
	** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia		** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia

Jet53man: /* Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses */

2021-07-27T22:44:16Z

Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses

← Older revision		Revision as of 18:44, 27 July 2021
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	==Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses==		==Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses==
	[[SSC/Structure/BiPolytropes/~~Analytic5_1~~#Limit_when_m3_.3D_0\|As we have shown elsewhere]], in the limit <math>~m_3 \rightarrow 0</math>, the physically relevant root of the above analytic relation is <math>~\xi_i = 3</math> and the mass contained in the core is,		[[SSC/Structure/BiPolytropes/Analytic15#Limit_when_m3_.3D_0\|As we have shown elsewhere]], in the limit <math>~m_3 \rightarrow 0</math>, the physically relevant root of the above analytic relation is <math>~\xi_i = 3</math> and the mass contained in the core is,
	<div align="center">		<div align="center">
	<math>		<math>
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	It should be clear from this analysis that the Bonnor-Ebert critical mass is not distinct from the Schönberg-Chandrasekhar critical mass. It can be derived from the Schönberg-Chandrasekhar mass in the limit when <math>\mu_e/\mu_c \rightarrow 0</math>. Had [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] examined their model in this limit, they would have "discovered" the Bonnor-Ebert mass a decade prior to both Bonnor's and Ebert's published works.		It should be clear from this analysis that the Bonnor-Ebert critical mass is not distinct from the Schönberg-Chandrasekhar critical mass. It can be derived from the Schönberg-Chandrasekhar mass in the limit when <math>\mu_e/\mu_c \rightarrow 0</math>. Had [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] examined their model in this limit, they would have "discovered" the Bonnor-Ebert mass a decade prior to both Bonnor's and Ebert's published works.

	{{~~LSU_WorkInProgress~~}}		{{ SGFworkInProgress }}

	<div align="center">		<div align="center">

Jet53man: /* Schönberg-Chandrasekhar Mass */

2021-07-27T22:43:01Z

Schönberg-Chandrasekhar Mass

← Older revision		Revision as of 18:43, 27 July 2021
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	In an effort to develop a more complete appreciation of the onset of the instability associated with the Schönberg-Chandrasekhar mass limit, [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Beech (1988)] matched an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an isothermal core and, like Schönberg & Chandrasekhar, allowed for a discontinuous change in the molecular weight at the interface. [For an even more comprehensive generalization and discussion, see [http://adsabs.harvard.edu/abs/2012MNRAS.421.2713B Ball, Tout, & Żytkow] (2012, MNRAS, 421, 2713)]. Beech's results were not significantly different from those reported by [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]; in particular, the value of <math>~\nu_\mathrm{max}</math> was still only definable numerically because an isothermal core cannot be described in terms of analytic functions.		In an effort to develop a more complete appreciation of the onset of the instability associated with the Schönberg-Chandrasekhar mass limit, [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Beech (1988)] matched an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an isothermal core and, like Schönberg & Chandrasekhar, allowed for a discontinuous change in the molecular weight at the interface. [For an even more comprehensive generalization and discussion, see [http://adsabs.harvard.edu/abs/2012MNRAS.421.2713B Ball, Tout, & Żytkow] (2012, MNRAS, 421, 2713)]. Beech's results were not significantly different from those reported by [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]; in particular, the value of <math>~\nu_\mathrm{max}</math> was still only definable numerically because an isothermal core cannot be described in terms of analytic functions.

	[[SSC/Structure/BiPolytropes/~~Analytic51~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|In an accompanying derivation]] [see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon] (1998, MNRAS, 298, 831)] we have gone one step farther, matching an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an analytically prescribable, <math>~n_c = 5</math> polytropic core. For this bipolytrope, we show that there is a limiting mass-fraction, <math>~\nu_\mathrm{max}</math>, for any choice of the molecular weight ratio <math>~\mu_c/\mu_e > 3</math> and that the interface location, <math>~\xi_i</math>, associated with this critical configuration is given by the positive, real root of the following relation:		[[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|In an accompanying derivation]] [see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon] (1998, MNRAS, 298, 831)] we have gone one step farther, matching an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an analytically prescribable, <math>~n_c = 5</math> polytropic core. For this bipolytrope, we show that there is a limiting mass-fraction, <math>~\nu_\mathrm{max}</math>, for any choice of the molecular weight ratio <math>~\mu_c/\mu_e > 3</math> and that the interface location, <math>~\xi_i</math>, associated with this critical configuration is given by the positive, real root of the following relation:

	<div align="center">		<div align="center">

Jet53man: /* Schönberg-Chandrasekhar Mass */

2021-07-27T22:41:14Z

Schönberg-Chandrasekhar Mass

← Older revision		Revision as of 18:41, 27 July 2021
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	In an effort to develop a more complete appreciation of the onset of the instability associated with the Schönberg-Chandrasekhar mass limit, [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Beech (1988)] matched an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an isothermal core and, like Schönberg & Chandrasekhar, allowed for a discontinuous change in the molecular weight at the interface. [For an even more comprehensive generalization and discussion, see [http://adsabs.harvard.edu/abs/2012MNRAS.421.2713B Ball, Tout, & Żytkow] (2012, MNRAS, 421, 2713)]. Beech's results were not significantly different from those reported by [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]; in particular, the value of <math>~\nu_\mathrm{max}</math> was still only definable numerically because an isothermal core cannot be described in terms of analytic functions.		In an effort to develop a more complete appreciation of the onset of the instability associated with the Schönberg-Chandrasekhar mass limit, [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Beech (1988)] matched an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an isothermal core and, like Schönberg & Chandrasekhar, allowed for a discontinuous change in the molecular weight at the interface. [For an even more comprehensive generalization and discussion, see [http://adsabs.harvard.edu/abs/2012MNRAS.421.2713B Ball, Tout, & Żytkow] (2012, MNRAS, 421, 2713)]. Beech's results were not significantly different from those reported by [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]; in particular, the value of <math>~\nu_\mathrm{max}</math> was still only definable numerically because an isothermal core cannot be described in terms of analytic functions.

	[[SSC/Structure/BiPolytropes/~~Analytic5_1~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|In an accompanying derivation]] [see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon] (1998, MNRAS, 298, 831)] we have gone one step farther, matching an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an analytically prescribable, <math>~n_c = 5</math> polytropic core. For this bipolytrope, we show that there is a limiting mass-fraction, <math>~\nu_\mathrm{max}</math>, for any choice of the molecular weight ratio <math>~\mu_c/\mu_e > 3</math> and that the interface location, <math>~\xi_i</math>, associated with this critical configuration is given by the positive, real root of the following relation:		[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|In an accompanying derivation]] [see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon] (1998, MNRAS, 298, 831)] we have gone one step farther, matching an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an analytically prescribable, <math>~n_c = 5</math> polytropic core. For this bipolytrope, we show that there is a limiting mass-fraction, <math>~\nu_\mathrm{max}</math>, for any choice of the molecular weight ratio <math>~\mu_c/\mu_e > 3</math> and that the interface location, <math>~\xi_i</math>, associated with this critical configuration is given by the positive, real root of the following relation:

	<div align="center">		<div align="center">

Jet53man: /* Bounded Isothermal Sphere & Bonnor-Ebert Mass */

2021-07-27T22:39:15Z

Bounded Isothermal Sphere & Bonnor-Ebert Mass

← Older revision		Revision as of 18:39, 27 July 2021
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	==Bounded Isothermal Sphere & Bonnor-Ebert Mass==		==Bounded Isothermal Sphere & Bonnor-Ebert Mass==
	In the mid-1950s, [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956) independently realized that an isothermal gas cloud can be stabilized by embedding it in a hot, tenuous external medium. The relevant mathematical model is constructed by chopping off the isothermal sphere at some finite radius — call it, <math>~\xi_e</math> — and imposing an externally applied pressure, <math>~P_e</math>, that is equal to the pressure of the isothermal gas at the specified edge of the truncated sphere. But for a given mass and temperature, there is a value of <math>~P_e</math> below which the truncated isothermal sphere is dynamically unstable, like its isolated and untruncated counterpart. Viewed another way, given the value of <math>~P_e</math> and the isothermal sound speed, <math>~c_s</math>, a bounded isothermal sphere will be dynamically stable only if its mass is below a critical value,		In the mid-1950s, [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956) independently realized that an isothermal gas cloud can be stabilized by embedding it in a hot, tenuous external medium. The relevant mathematical model is constructed by chopping off the isothermal sphere at some finite radius — call it, <math>\xi_e</math> — and imposing an externally applied pressure, <math>~P_e</math>, that is equal to the pressure of the isothermal gas at the specified edge of the truncated sphere. But for a given mass and temperature, there is a value of <math>P_e</math> below which the truncated isothermal sphere is dynamically unstable, like its isolated and untruncated counterpart. Viewed another way, given the value of <math>P_e</math> and the isothermal sound speed, <math>c_s</math>, a bounded isothermal sphere will be dynamically stable only if its mass is below a critical value,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" width="95%">		<table border="0" cellpadding="5" width="95%">
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	<th align="center">		<th align="center">
	<font size="+1">		<font size="+1">
	<math>~\alpha</math>		<math>\alpha</math>
	</font>		</font>
	</th>		</th>
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	<tr>		<tr>
	<td align="center">		<td align="center">
	<math>~1.18</math>		<math>1.18</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="center">		<td align="center">
	[[~~SphericallySymmetricConfigurations~~/Virial#Bounded_Isothermal\|Here]]		[[SSCpt1/Virial#Bounded_Isothermal\|Here]]
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	[[~~SphericallySymmetricConfigurations~~/Virial#Bounded_Adiabatic\|Here]]		[[SSCpt1/Virial#Bounded_Adiabatic\|Here]]
	</td>		</td>
	</tr>		</tr>
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	where <math>~\alpha</math> is a dimensionless coefficient of order unity. This limiting mass is often referred to as the Bonnor-Ebert mass. It appears most frequently in the astrophysics literature in discussions of star formation because that is the arena in which both Bonnor and Ebert were conducting research when they made their discoveries.		where <math>~\alpha</math> is a dimensionless coefficient of order unity. This limiting mass is often referred to as the Bonnor-Ebert mass. It appears most frequently in the astrophysics literature in discussions of star formation because that is the arena in which both Bonnor and Ebert were conducting research when they made their discoveries.

	As is [[SSC/Structure/BonnorEbert#Maximum_Mass\|reviewed in a related discussion]] and as is documented in the table accompanying the expression for <math>~M_\mathrm{max}</math>, above, [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956) used [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Emden's (1907) tabulated properties]] of an isothermal sphere to determine that the dimensionless radius of this limiting configuration is <math>\xi_e \approx 6.5</math> and that the leading coefficient, <math>\alpha \approx 1.18</math>. It is worth noting that a [[~~SphericallySymmetricConfigurations~~/Virial#BonnorEbertMass\|global virial analysis of the stability of bounded isothermal spheres]] produces the same expression for <math>~M_\mathrm{max}</math> with a leading coefficient that has an exact, analytic prescription, namely, <math>\alpha = (3^4 \cdot 5^3/2^{10}\pi)^{1/2} \approx 1.77408</math>. While it can be advantageous to reference this analytic prescription of <math>\alpha</math>, the virial analysis must be considered more approximate than Bonnor's analysis because it does not require the construction of models that are in detailed force balance.		As is [[SSC/Structure/BonnorEbert#Maximum_Mass\|reviewed in a related discussion]] and as is documented in the table accompanying the expression for <math>~M_\mathrm{max}</math>, above, [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956) used [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Emden's (1907) tabulated properties]] of an isothermal sphere to determine that the dimensionless radius of this limiting configuration is <math>\xi_e \approx 6.5</math> and that the leading coefficient, <math>\alpha \approx 1.18</math>. It is worth noting that a [[SSCpt1/Virial#BonnorEbertMass\|global virial analysis of the stability of bounded isothermal spheres]] produces the same expression for <math>M_\mathrm{max}</math> with a leading coefficient that has an exact, analytic prescription, namely, <math>\alpha = (3^4 \cdot 5^3/2^{10}\pi)^{1/2} \approx 1.77408</math>. While it can be advantageous to reference this analytic prescription of <math>\alpha</math>, the virial analysis must be considered more approximate than Bonnor's analysis because it does not require the construction of models that are in detailed force balance.

	Our [[SSC/Structure/PolytropesEmbedded#Extension_to_Bounded_Sphere_2\|detailed force-balance analysis of truncated and pressure-bounded, <math>~n=5</math> polytropes]] identifies a physically analogous limiting mass. If the average isothermal sound speed, <math>~\bar{c_s}</math>, [[~~SphericallySymmetricConfigurations~~/Virial#average\|as defined elsewhere]], is used in place of <math>c_s</math>, the mathematical expression for <math>~M_\mathrm{max}</math> has exactly the same form as in the isothermal case. But for the <math>~n=5</math> polytrope we know that the limiting configuration has a dimensionless radius given precisely by <math>~\xi_e = 3</math>; and, as a result, the leading coefficient in the definition of <math>~M_\mathrm{max}</math> is prescribable analytically, namely, <math>~\alpha = 2^{-3/10} \cdot (3^7/2^8 \pi)^{1/2} \approx 1.33943</math>. As is documented in the table accompanying the relation for <math>~M_\mathrm{max}</math>, above, in the case of a truncated <math>~n=5</math> polytrope, [[~~SphericallySymmetricConfigurations~~/Virial#Maximum_Mass\|the simpler and more approximate virial analysis gives]], <math>~\alpha = (3^{19}/2^{12}\cdot 5^7\pi)^{1/2} \approx 1.07523</math>.		Our [[SSC/Structure/PolytropesEmbedded#Extension_to_Bounded_Sphere_2\|detailed force-balance analysis of truncated and pressure-bounded, <math>n=5</math> polytropes]] identifies a physically analogous limiting mass. If the average isothermal sound speed, <math>~\bar{c_s}</math>, [[SSCpt1/Virial#average\|as defined elsewhere]], is used in place of <math>c_s</math>, the mathematical expression for <math>~M_\mathrm{max}</math> has exactly the same form as in the isothermal case. But for the <math>~n=5</math> polytrope we know that the limiting configuration has a dimensionless radius given precisely by <math>\xi_e = 3</math>; and, as a result, the leading coefficient in the definition of <math>M_\mathrm{max}</math> is prescribable analytically, namely, <math>\alpha = 2^{-3/10} \cdot (3^7/2^8 \pi)^{1/2} \approx 1.33943</math>. As is documented in the table accompanying the relation for <math>~M_\mathrm{max}</math>, above, in the case of a truncated <math>~n=5</math> polytrope, [[SSCpt1/Virial#Maximum_Mass\|the simpler and more approximate virial analysis gives]], <math>\alpha = (3^{19}/2^{12}\cdot 5^7\pi)^{1/2} \approx 1.07523</math>.

	==Schönberg-Chandrasekhar Mass==		==Schönberg-Chandrasekhar Mass==

Jet53man at 22:35, 27 July 2021

2021-07-27T22:35:32Z

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Jet53man: /* Related Discussions */

2021-07-27T22:31:54Z

Related Discussions

← Older revision		Revision as of 18:31, 27 July 2021
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	* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]		* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]
	* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]		* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]
	* [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]		* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]
	* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]		* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]
	** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia		** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia

← Older revision		Revision as of 18:44, 27 July 2021
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	* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]		* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]
	* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]		* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]
	* [[SSC/Structure/BiPolytropes/~~Analytic51~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]		* [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]
	* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]		* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]
	** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia		** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia

← Older revision		Revision as of 18:31, 27 July 2021
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	* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]		* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]
	* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]		* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]
	* [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]		* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]
	* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]		* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]
	** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia		** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia